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Second Grade Unit Two

Georgia Standards of ExcellenceCurriculum Frameworks

GSE Second GradeUnit 2: Becoming Fluent with Addition and Subtraction

Georgia Department of Education

Georgia Standards of Excellence Framework

GSE Becoming Fluent with Addition and Subtraction Unit 2

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MATHEMATICS GSE Grade 2 Unit #2:Becoming Fluent with Addition and Subtraction

Richard Woods, State School Superintendent

July 2017 Page 131 of 132

All Rights Reserved

Unit Two: Becoming Fluent with Addition and Subtraction

TABLE OF CONTENTS

Overview....

3

Standards for Mathematical Practice.

5

Standards for Mathematical Content .

6

Big Ideas ....

7

Essential Questions

7

Concepts and Skills to Maintain

8

Strategies for Teaching and Learning

9

Selected Terms and Symbols .....

18

Task Table .

20

Intervention Table 23

Got Milk? ..24

Incredible Equations .31

Order is Important ....35

Different Paths, Same Destination 40

Number Destinations 46

Our Number Riddles/My Number Riddle ..51

Building/Busting Towers of 10 .57

Story Problems ..61

Roll Away .....65

Mental Math ..71

Take 100 ....77

Multi-digit Addition ..83

Addition Strategies ....88

Caterpillars and Leaves (FAL) ..93

Sale Flyer Shopping ..94

Grocery Store Math ...99

Subtraction: Modeling with Regrouping 102

Subtraction Story Problems ...109

Menu Math ....113

Counting Mice ...117

Every Picture Tells a Story ....123

Planning a Field Trip .126

IF YOU HAVE NOT READ THE 2nd GRADE CURRICULUM OVERVIEW IN ITS ENTIRETY PRIOR TO USE OF THIS UNIT, PLEASE STOP AND CLICK HERE: https://www.georgiastandards.org/Georgia-Standards/Frameworks/2nd-Math-Grade-Level-Overview.pdf

Return to the use of this unit once youve completed reading the Curriculum Overview. Thank you.

OVERVIEW

In this unit students will:

cultivate an understanding of how addition and subtraction affect quantities and are related to each other

will reinforce the multiple meanings for addition (combine, join, and count on) and subtraction (take away, remove, count back, and compare)

further develop their understanding of the relationships between addition and subtraction

recognize how the digits 0-9 are used in our place value system to create numbers and manipulate amounts

continue to develop their understanding solving problems with money

At the beginning of Unit 2, it is recommended that students practice counting money collections daily during Number Corner or as part of daily Math Maintenance in order to be prepared for the tasks. For more detailed information regarding unpacking the content standards, unpacking a task, math routines and rituals, maintenance activities and more, please refer to the grade level overview.

As students in second grade begin to count larger amounts, they should group concrete materials into tens and ones to keep track of what they have counted. This is the introduction of our place value system where students must learn that the digits (0-9) have different values depending on their position in a number.

Students in second grade now build on their work with one-step problems to solve two-step problems. Second graders need to model and solve problems and represent their solutions with equations. The problems should involve sums and differences less than or equal to 100 using the numbers 0 to 100. Picture Graphs and Bar Graphs are also introduced in second grade. Investigations and experiences with graphing should take place all year long.

Addition and Subtraction in Elementary School

(Information adapted from North Carolina DPI Instructional Support Tools)

The strategies that students use to solve problems provide important information concerning number sense, and place value.

It is important to look at more than answers students get. The strategies used provide useful information about what problems to give the next day, and how to differentiate instruction.

It is important to relate addition and subtraction.

Student-created strategies provide reinforcement of place value concepts. Traditional algorithms can actually unteach place value.

Student created strategies are built on a students actual understanding, instead of on what the book says or what we think/hope they know!

Students make fewer errors with invented strategies, because they are built on understanding rather than memorization.

Students use various counting strategies, including counting all, counting on, and counting back with numbers up to 20. This standard calls for students to move beyond counting all and become comfortable at counting on and counting back. The counting all strategy requires students to count an entire set. The counting and counting back strategies occur when students are able to hold the start number in their head and count on from that number.

Although the units in this instructional framework emphasize key standards and big ideas at specific times of the year, routine topics such as counting, time, money, positional words, patterns, and tallying should be addressed on an ongoing basis. Additionally, the required fluency expectations for second grade students (knowing from memory all sums of two one digit numbers) should be a gradual progression. The word fluency is used judiciously in the standards to mark the endpoints of progressions of learning that begin with solid underpinnings and then pass upward through stages of growing maturity. By doing this we are allowing students to gradually enhance their understanding of the concept of number and to develop computational proficiency.

NUMBER TALKS

Between 5 and 15 minutes each day should be dedicated to Number Talks in order to build students mental math capabilities and reasoning skills. Sherry Parrishs book Number Talks provides examples of K-5 number talks. The following video clip from Math Solutions is an excellent example of a number talk in action. http://www.mathsolutions.com/videopage/videos/Final/Classroom_NumberTalk_Gr3.swf

During the Number Talk, the teacher is not the definitive authority. The teacher is the facilitator and is listening for and building on the students natural mathematical thinking. The teacher writes a problem horizontally on the board in whole group or a small setting. The students mentally solve the problem and share with the whole group how they derived the answer. They must justify and defend their reasoning. The teacher simply records the students thinking and poses extended questions to draw out deeper understanding for all.

The effectiveness of Numbers Talks depends on the routines and environment that is established by the teacher. Students must be given time to think quietly without pressure from their peers. To develop this, the teacher should establish a signal, other than a raised hand, of some sort to identify that one has a strategy to share. One way to do this is to place a finger on their chest indicating that they have one strategy to share. If they have two strategies to share, they place out two fingers on their chest and so on.

Number Talk problem possible student responses:

Possible Strategy #1

Possible Strategy #2

29 + 8

29 can become 30 and

Take 1 from 8 reducing it to 7.

9 and 8 becomes 17

17 plus 20

54 + 86

50 + 80 + 10 =

Add 6 to 54 to get 60.

Then 60 + 80 = 140

Number talks often have a focus strategy such as making tens or compensation. Providing students with a string of related problems, allows students to apply a strategy from a previous problem to subsequent problems. Some units lend themselves well to certain Number Talk topics. For example, the place value unit may coordinate well with the Number Talk strategy of making ten.

Note: Most of the concepts in this unit can be supported through 10-minute daily Number Talks. If you are doing effective daily Number Talks, your students may move more quickly through this unit and/or provide you with excellent formative information regarding your future daily Number Talks and implementation of the tasks.

STANDARDS FOR MATHEMATICAL PRACTICE

This section provides examples of learning experiences for this unit that support the development of the proficiencies described in the Standards for Mathematical Practice. The statements provided offer a few examples of connections between the Standards for Mathematical Practice and the Content Standards of this unit. The list is not exhaustive and will hopefully prompt further reflection and discussion.

1. Make sense of problems and persevere in solving them. Students have multiple opportunities to develop strategies for mental math addition and subtraction as well as solving story problems, riddles, and graphs.

2. Reason abstractly and quantitatively. Students use number lines, base ten blocks, money, and other manipulatives to connect quantities to written symbols. Students compare numbers and discover the commutative property of addition.

3. Construct viable arguments and critique the reasoning of others. Students develop strategies for mental math as well as solving story problems and interpreting graphs. The students share and defend their thinking.

4. Model with mathematics. Students use words, pictures, graphs, money, and manipulatives to express addition and subtraction problems.

5. Use appropriate tools strategically. Students use estimation, pictures, and manipulatives to solve addition and subtraction computation as well as story problems.

6. Attend to precision. Students have daily practice in number talks and tasks to use mathematical language to explain their own reasoning.

7. Look for and make use of structure. Students look for patterns developing mental strategies: making tens, repeated addition, fact families, and doubles.

8. Look for and express regularity in repeated reasoning. Students look for shortcuts in addition mental math: such as rounding up, then adjusting; repeated addition; riddles; and reasonableness of answers.

***Mathematical Practices 1 and 6 should be evident in EVERY lesson. ***

STANDARDS FOR MATHEMATICAL CONTENT

Represent and solve problems involving addition and subtraction.

MGSE2.OA.1 Use addition and subtraction within 100 to solve one and two step word problems by using drawings and equations with a symbol for the unknown number to represent the problem. Problems include contexts that involve adding to, taking from, putting together/taking apart (part/part/whole) and comparing with unknowns in all positions.

Add and subtract within 20.

MGSE2.OA.2 Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Use place value understanding and properties of operations to add and subtract.

MGSE2.NBT.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Work with time and money.

MGSE2.MD.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?

Represent and interpret data

MGSE2.MD.10 Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems[footnoteRef:1] using information presented in a bar graph [1: ]

BIG IDEAS

By the conclusion of this unit, students should be able to demonstrate the following competencies:


Solve a variety of word problems involving money using $ and symbols.

Understand and apply properties of operations and the relationship between addition and subtraction.

Recognize how the digits 0-9 are used in our place value system to create numbers and manipulate amounts.

Understand how addition and subtraction affect quantities and are related to each other.

Know the multiple meanings for addition (combine, join, and count on) and subtraction (take away, remove, count back, and compare)

Use the inverse operation to check that they have correctly solved the problem.

Solve problems using mental math strategies.

Draw and interpret picture and bar graphs to represent a data set with up to four categories.

Essential Questions

How do we represent a collection of objects using tens and ones?

How do I express money amounts?

When will estimating be helpful to us?

How can we use skip counting to help us solve problems?

Can we change the order of numbers if we subtract? Why or why not?

Can we change the order of numbers when we add (or subtract)? Why or why not?

How can estimation strategies help us build our addition skills?

How do we use addition to tell number stories?

How can benchmark numbers help us add?

How does using ten as a benchmark number help us add and subtract?

What strategies can help us when adding and subtracting with regrouping?

What strategies will help me add multiple numbers quickly and accurately?

How can we solve addition problems with and without regrouping?

How can addition help us know we subtracted two numbers correctly?

How can we solve subtraction problems with and without regrouping?

How can strategies help us when adding and subtracting with regrouping?

How can we model and solve subtraction problems with and without regrouping? How can mental math strategies, for example estimation and benchmark numbers, help us when adding and subtracting with regrouping?

How can I use a number line to help me model how I combine and compare numbers?

How are addition and subtraction alike and how are they different?

What is a number sentence and how can I use it to solve word problems?

How do we solve problems in different ways?

How can we solve problems mentally? What strategies help us with this?

How can we show/represent problems in different ways?

How can problem situations and problem-solving strategies be represented?

How are problem-solving strategies alike and different?

How can different combinations of numbers and operations be used to represent the same quantity?

CONCEPTS AND SKILLS TO MAINTAIN

Fluency: Procedural fluency is defined as skill in carrying out procedures flexibly, accurately, efficiently, and appropriately. Fluent problem solving does not necessarily mean solving problems within a certain time limit, though there are reasonable limits on how long computation should take. Fluency is based on a deep understanding of quantity and number.

Deep Understanding: Teachers teach more than simply how to get the answer and instead support students ability to access concepts from a number of perspectives. Therefore, students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of foundational mathematics concepts by applying them to new situations, as well as writing and speaking about their understanding.

Memorization: The rapid recall of arithmetic facts or mathematical procedures. Memorization is often confused with fluency. Fluency implies a much richer kind of mathematical knowledge and experience.

Number Sense: Students consider the context of a problem, look at the numbers in a problem, and make a decision about which strategy would be most efficient in each particular problem. Number sense is not a deep understanding of a single strategy, but rather the ability to think flexibly between varieties of strategies in context.

Fluent students:

flexibly use a combination of deep understanding, number sense, and memorization.

are fluent in the necessary baseline functions in mathematics so that they are able to spend their thinking and processing time unpacking problems and making meaning from them.

are able to articulate their reasoning.

find solutions through a number of different paths.

For more about fluency, see:

http://www.youcubed.org/wp-content/uploads/2015/03/FluencyWithoutFear-2015.pdf and:

https://bhi61nm2cr3mkdgk1dtaov18-wpengine.netdna-ssl.com/wp-content/uploads/nctm-timed-tests.pdf

Skills from Grade 1:

It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas.

Developing understanding of addition, subtraction, and strategies for addition and subtraction within 20;

Developing understanding of whole number relationships and place value, including grouping in tens and ones;

Second Grade Year Long Concepts:

Organizing and graphing data as stated in MGSE2.MD.10 should be regularly incorporated in activities throughout the year. Students should be able to draw a picture graph and a bar graph to represent a data set with up to four categories as well as solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Routine topics such as counting, time, money, positional words, patterns, and tallying should be addressed on an ongoing basis throughout instructional time. These topics that should also be addressed daily through Number Corner or Math Maintenance.

Students will be asked to use estimation and benchmark numbers throughout the year in a variety of mathematical situations.

STRATEGIES FOR TEACHING AND LEARNING

(Information adapted from North Carolina DPI Instructional Support Tools)


MGSE2.OA.1 Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Instructional Strategies

This standard calls for students to add and subtract numbers within 100 in the context of one and two step word problems. Students should have ample experiences working on various types of problems that have unknowns in all positions, including Result Unknown, Change Unknown, and Start Unknown. See Table 1 on page 11 for further examples.

The problems should involve sums and differences less than or equal to 100 using the numbers 0 to 100. It is vital that students develop the habit of checking their answer to a problem to determine if it makes sense for the situation and the questions being asked.

This standard also calls for students to solve one- and two-step problems using drawings, objects and equations. Students can use place value blocks or hundreds charts, or create drawings of place value blocks or number lines to support their work. Examples of one-step problems with unknowns in different places are provided in Table 1. Two step-problems include situations where students have to add and subtract within the same problem.

Example:

In the morning there are 25 students in the cafeteria. 18 more students come in. After a few minutes, some students leave. If there are 14 students still in the cafeteria, how many students left the cafeteria? Write an equation for your problem.

Student 1

Step 1

I used place value blocks and made a group of 25 and a group of 18. When I counted them I had 3 tens and 13 ones which is 43.

Step 2

I then wanted to remove blocks until there were only 14 left. I removed blocks until there were 20 left.

Step 3

Since I have two tens I need to trade a ten for 10 ones.

Step 4

After I traded it, I removed blocks until there were only 14 remaining.

Step 5

My answer was the number of blocks that I removed. I removed 2 tens and 9 ones. Thats 29.

My equation is 25 + 18 ___ = 14.

Student 2

I used a number line. I started at 25 and needed to move up 18 spots so I started by moving up 5 spots to 30, and then 10 spots to 40, and then 3 more spots to 43. Then I had to move backwards until I got to 14 so I started by first moving back 20 spots until I got to 23. Then I moved to 14 which were an additional 9 places. I moved back a total of 29 spots. Therefore, there were a total of 29 students left in the cafeteria.

My equation is: 25 + 18 ___ = 14.

Student 3

Step 1

I used a hundreds board. I started at 25. I moved down one row which is 10 more, then moved to the right 8 spots and landed on 43. This represented the 18 more students coming into the cafeteria.

Step 2

Now starting at 43, I know I have to get to the number 14 which represents the number of students left in the cafeteria so I moved up 2 rows to 23 which is 20 less. Then I moved to the left until I land on 14, which is 9 spaces. I moved back a total of 29 spots. That means 29 students left the cafeteria.

Step 3

My equation to represent this situation is 25 + 18 ___ = 14.

Working on addition and subtraction simultaneously, continually relating the two operations is important for helping students recognize and understand the (inverse) relationship of these two operations. It is also vital that students develop the habit of checking their answer to a problem to determine if it makes sense for the situation and the questions being asked. An excellent way to do this is to ask students to write word problems for their classmates to solve. A good place to start is by giving students the answer to a problem. Then tell students whether you want them to write an addition or subtraction problem situation. Also let them know that the sums and differences can be less than or equal to 100. For example, ask students to write an addition word problem for their classmates to solve which requires adding four two-digit numbers with 100 as the answer. Students then share, discuss and compare their solution strategies after they solve the problems.

The strategies that students use to solve problems provide important information concerning number sense and place value understandings therefore it is important to look at more than answers students get. The strategies students use provide useful information about what problems to give the next day and how to differentiate instruction. Student-created strategies provide reinforcement of place value concepts. Teaching traditional algorithms can actually hinder the development of conceptual knowledge of our place value system; whereas student created strategies are built on a students actual understanding, instead of on what the book says or what we think/hope they know. Students make fewer errors with their own invented strategies because they are built on their own understanding rather than memorization.

Table 1: Common addition and subtraction situations

Result Unknown

Change Unknown

Start Unknown

Add to

Two bunnies sat on the grass.

Three more bunnies hopped

there. How many bunnies are

on the grass now?

2 + 3 = ?

Two bunnies were sitting

on the grass. Some more

bunnies hopped there. Then

there were five bunnies. How

many bunnies hopped over

to the first two?

2 + ? = 5

Some bunnies were sitting

on the grass. Three more

bunnies hopped there. Then

there were five bunnies. How

many bunnies were on the

grass before?

? + 3 = 5

Take from

Five apples were on the table. I ate two apples. How many apples are on the table now?

5 2 = ?

Five apples were on the table. I ate some apples. Then there were three apples. How many apples did I eat?

5 ? = 3

Some apples were on the

table. I ate two apples. Then

there were three apples. How

many apples were on the table before?

? 2 = 3

Total Unknown

Addend Unknown

Both Addends Unknown

Put Together

Take Apart

Three red apples and two green apples are on the table. How many apples are on the table?

3 + 2 = ?

Five apples are on the table.

Three are red and the rest are green. How many apples are green?

3 + ? = 5, 5 3 = ?

Grandma has five flowers.

How many can she put in her red vase and how many in her blue vase?

5 = 0 + 5, 5 = 5 + 0

5 = 1 + 4, 5 = 4 + 1

5 = 2 + 3, 5 = 3 + 2

Difference Unknown

Bigger Unknown

Smaller Unknown

Compare

(How many more? version):

Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy?

(How many fewer? version):

Lucy has two apples. Julie has five apples. How many fewer apples does Lucy have than Julie?

2 + ? = 5, 5 2 = ?

(Version with more):

Julie has three more apples than Lucy. Lucy has two apples. How many apples does Julie have?

(Version with fewer):

Lucy has 3 fewer apples than

Julie. Lucy has two apples.

How many apples does Julie have?

2 + 3 = ?, 3 + 2 = ?

(Version with more):

Julie has three more apples

than Lucy. Julie has five

apples. How many apples

does Lucy have?

(Version with fewer):

Lucy has 3 fewer apples than

Julie. Julie has five apples.

How many apples does Lucy have?

5 3 = ?, ? + 3 = 5

Adapted from Box 2-4 of Mathematics Learning in Early Childhood, National Research Council (2009, pp. 32, 33).

Add and Subtract within 20.



This standard mentions the word fluently when students are adding and subtracting numbers within 20. Fluency means accuracy (correct answer), efficiency (within 4-5 seconds), and flexibility (using strategies such as making 10 or breaking apart numbers). Research indicates that teachers can best support students memorization of sums and differences through varied experiences making 10, breaking numbers apart and working on mental strategies, rather than repetitive timed tests.

Example: 9 + 5 = ___

Student 1: Counting On

I started at 9 and then counted 5 more. I landed at 14.

Student 2: Decomposing a Number Leading to a Ten

I know that 9 and 1 is 10, so I broke 5 into 1 and 4. 9 plus 1 is 10. Then I have to add 4 more, which gets me to 14.

Example: 13 9 = ___

Student 1: Using the Relationship between Addition and Subtraction

I know that 9 plus 4 equals 13. So 13 minus 9 equals 4.

Student 2: Creating an Easier Problem

I added 1 to each of the numbers to make the problem 14 minus 10. I know the answer is 4. So 13 minus 9 is also 4.

Student 3: Using the benchmark of 10

I know that 13 minus 3 equals 10, so I take 3 away from the 9 and 3 away from the 13. 10 minus 6 equals 4.

Provide many activities that will help students develop a strong understanding of number relationships, addition and subtraction so they can develop, share and use efficient strategies for mental computation. An efficient strategy is one that can be done mentally and quickly. Students gain computational fluency, using efficient and accurate methods for computing, as they come to understand the role and meaning of arithmetic operations in number systems. Efficient mental processes become automatic with use.

Provide activities in which students apply the commutative and associative properties to their mental strategies for sums less or equal to 20 using the numbers 0 to 20.

Have students study how numbers are related to 5 and 10 so they can apply these relationships to their strategies for knowing 5 + 4 or 8 + 3. Students might picture 5 + 4 on a ten-frame to mentally see 9 as the answer. For remembering 8 + 7, students might think: since 8 is 2 away from 10, take 2 away from 7 to make 10 + 5 = 15. Activities such as these will provide good opportunities to use number talks as described in the 2nd Grade Overview. Example: When presented the problem, 4 + 8 + 6, the student uses number talk to say I know 6 + 4 = 10, so I can add 4 + 8 + 6 by adding 4 + 6 to make 10 and then add 8 to make 18. Make anchor charts/posters for student-developed mental strategies for addition and subtraction within 20. Use names for the strategies that make sense to the students and include examples of the strategies. Present a particular strategy along with the specific addition and subtraction facts relevant to the strategy. Have students use objects and drawings to explore how these facts are alike.

Provide simple word problems designed for students to invent and try a particular strategy as they solve it. Have students explain their strategies so that their classmates can understand it. Guide the discussion so that the focus is on the methods that are most useful. Encourage students to try the strategies that were shared so they can eventually adopt efficient strategies that work for them. Use anchor charts/posters illustrating the various student strategies to use as reference as the students develop their toolbox of strategies.

Use place value understanding and properties of operations to add and subtract.

MGSE2.NBT.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. (For additional information please see the Grade Level Overview)

**This standard mentions the word fluently, just as stated with MGSE2.OA.2, when students are adding and subtracting numbers within 100. Fluency means accuracy (correct answer), efficiency (basic facts computed within 4-5 seconds), and flexibility (using strategies such as making 10 or decomposing).

This standard calls for students to use pictorial representations and/or strategies to find the solution. Students who are struggling may benefit from further work with concrete objects (e.g., place value blocks).

Provide many activities that will help students develop a strong understanding of number relationships, addition and subtraction so they can develop, share and use efficient strategies for mental computation. An efficient strategy is one that can be done mentally and quickly. Students gain computational fluency, using efficient and accurate methods for computing, as they come to understand the role and meaning of arithmetic operations in number systems. Efficient mental processes become automatic with use.

Students need to build on their flexible strategies for adding within 100 in Grade 1 to fluently add and subtract within 100, add up to four two-digit numbers, and find sums and differences less than or equal to 1000 using numbers 0 to 1000.

Initially, students apply base-ten concepts and use direct modeling with physical objects or drawings to find different ways to solve problems. They move to inventing strategies that do not involve physical materials or counting by ones to solve problems. Student-invented strategies likely will be based on place-value concepts, the commutative and associative properties, and the relationship between addition and subtraction. These strategies should be done mentally or with a written record for support.

It is vital that student-invented strategies be shared, explored, recorded and tried by others. Recording the expressions and equations in the strategies horizontally encourages students to think about the numbers and the quantities they represent instead of the digits. Not every student will invent strategies, but all students can and will try strategies they have seen that make sense to them. Different students will prefer different strategies.

Students will decompose and compose tens and hundreds when they develop their own strategies for solving problems where regrouping is necessary. They might use the make-ten strategy (37 + 8 = 40 + 5 = 45, add 3 to 37 then 5) or (62 - 9 = 60 7 = 53, take off 2 to get 60, then 7 more) because no ones are exchanged for a ten or a ten for ones.

Have students analyze problems before they solve them. Present a variety of subtraction problems within 1000. Ask students to identify the problems requiring them to decompose the tens or hundreds to find a solution and explain their reasoning.

Example: 67 + 25 = __

Place Value Strategy

I broke both 67 and 25 into tens and ones. 6 tens plus 2 tens equals 8 tens. Then I added the ones. 7 ones plus 5 ones equals 12 ones. I then combined my tens and ones. 8 tens plus 12 ones equals 92.

Counting On and Decomposing a Number Leading to Ten

I wanted to start with 67 and then break 25 apart. I started with 67 and counted on to my next ten. 67 plus 3 gets me to 70. Then I added 2 more to get to 72. I then added my 20 and got to 92.

Commutative Property

I broke 67 and 25 into tens and ones so I had to add 60 + 7 + 20 + 5. I added 60 and 20 first to get 80. Then I added 7 to get 87. Then I added 5 more. My answer is 92.

Example: 63 32 = __

Relationship between Addition and Subtraction

I broke apart both 63 and 32 into tens and ones. I know that 2 plus 1 equals 3, so I have 1 left in the ones place. I know that 3 plus 3 equals 6, so I have a 3 in my tens place. My answer has a 1 in the ones place and 3 in the tens place, so my answer is 31.

Work with Money (For additional information please refer to the Grade Level Overview)

MGSE2.MD.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?

Relate to whole-number place value and base-ten understandings. For example, 23 = 2 dimes and 3 pennies.

Understand the relationship between quantity and value. For example, 1 dime = 10. Help students to understand that the relationship between coin size and value is inconsistent.

Limit problems to the use of just dollar and cents symbols. There should be no decimal notation for money at this point.

This standard calls for students to solve word problems involving either dollars or cents. Since students have not been introduced to decimals, problems should either have only dollars or only cents.

Example: What are some possible combinations of coins (pennies, nickels, dimes, and quarters) that equal 37 cents?

Example: What are some possible combinations of dollar bills ($1, $5 and $10) that equal 12 dollars?

The topic of money begins at Grade 2 and builds on the work in other clusters in this and previous grades. Help students learn money concepts and solidify their understanding of other topics by providing activities where students make connections between them. For instance, link the value of a dollar bill as 100 cents to the concept of 100 and counting within 1000. Use play money - nickels, dimes, and dollar bills to skip count by 5s, 10s, and 100s. Reinforce place value concepts with the values of dollar bills, dimes, and pennies. Students use the context of money to find sums and differences less than or equal to 100 using the numbers 0 to 100. They add and subtract to solve one- and two-step word problems involving money situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions. Students use drawings and equations with a symbol for the unknown number to represent the problem. The dollar sign, $, is used for labeling whole-dollar amounts without decimals, such as $29. Students need to learn the relationships between the values of a penny, nickel, dime, quarter and dollar bill.

Represent and Interpret Data

MGSE2.MD.10 Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.


At first students should create real object and picture graphs so each row or bar consists of countable parts. These graphs show items in a category and do not have a numerical scale. For example, a real object graph could show the students shoes (one shoe per student) lined end to end in horizontal or vertical rows by their color. Students would simply count to find how many shoes are in each row or bar. The graphs should be limited to 2 to 4 rows or bars. Students would then move to making horizontal or vertical bar graphs with two to four categories and a single-unit scale.

Flavor

Number of People

Chocolate

12

Vanilla

5

Strawberry

6

Cherry

9

Students display their data using a picture graph or bar graph using a single unit scale.

Favorite Ice Cream Flavor

Chocolate

Vanilla

Strawberry

Cherry

1 2 3 4 5 6 7 8 9 10 11

As students continue to develop their use of reading and interpreting data, it is highly suggested to incorporate these standards into daily routines. It is not merely the making or filling out of the graph, but the connections made from the date represented that builds and strengthens mathematical reasoning.

Use the information in the graphs to pose and solve simple put together, take-apart, and compare problems illustrated in Table 1 located on page 12.

SELECTED TERMS AND SYMBOLS

The following terms and symbols are not an inclusive list and should not be taught in isolation. Instructors should pay particular attention to them and how their students are able to explain and apply them (i.e. students should not be told to memorize these terms).

Teachers should present these concepts to students with models and real-life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers.

For specific definitions, please reference the Georgia Standards of Excellence State Standards Glossary.

add

addition and subtraction within 5, 10, 20, 100, or 1000

associative property for addition

bar graph

commutative property for addition

comparing

counting strategy

difference

doubles plus one

equations

estimating: fluency

fluently

identity property for addition

join

line plot

picture graph

place value

quantity

recalling facts

re-grouping

remove

scale

strategies for addition

strategy

subtract

unknowns

Link to Visual Vocabulary Cards: http://www.ncesd.org/Page/983

TASKS

The following tasks represent the level of depth, rigor, and complexity expected of all second grade students. These tasks or a task of similar depth and rigor should be used to demonstrate evidence of learning. It is important that all elements of a task be addressed throughout the learning process so that students understand what is expected of them.

To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important that the tasks be reviewed prior to instruction. The tasks in this unit illustrate the types of learning activities that should be conducted to meet the GSE. A variety of additional resources should be utilized to supplement these tasks.

TASK TYPES

Scaffolding Task

Tasks that build up to the learning task.

Constructing Task

Constructing understanding through deep/rich contextualized problem solving tasks.

Practice Task

Tasks that provide students opportunities to practice skills and concepts.

Culminating Task

Designed to require students to use several concepts learned during the unit to answer a new or unique situation. Allows students to give evidence of their own understanding toward the mastery of the standard and requires them to extend their chain of mathematical reasoning.

Formative Assessment Lesson (FAL)

Lessons that support teachers in formative assessment which both reveal and develop students understanding of key mathematical ideas and applications. These lessons enable teachers and students to monitor in more detail their progress towards the targets of the standards.

3-Act Task

A Three-Act Task is a whole-group mathematics task consisting of 3 distinct parts: an engaging and perplexing Act One, an information and solution seeking Act Two, and a solution discussion and solution revealing Act Three. More information along with guidelines for 3-Act Tasks may be found in the Guide to Three-Act Tasks on georgiastandards.org and the K-5 GSE Mathematics Wiki.

Task Name

Task Type/Grouping

Strategy

Content Addressed

Standard(s)

Task Description

Students will:

Look ahead to the FORMATIVE ASSESSMENT Lesson (FAL) listed in the table below. You may consider giving the FAL at the beginning of the unit as a pre-assessment. The official administration of the FAL takes place approximately 2/3 of the way through the unit.

You will design your Formative Assessment Lesson based on the data you collect from giving the Formative Assessment provided.

Got Milk

3 Act Task

Adding and Subtracting within 100

MGSE2.OA.1

MGSE2.NBT.5

Practice adding and subtracting two-digit numbers with explanations in number, picture, and word form.

Incredible Equations

Scaffolding Task

Large Group, Small Groups

Composing and decomposing numbers

MGSE2.OA.2

Gain additional understanding of the magnitude of a given number as well as its relationship to other numbers; learn multiple ways of composing decomposing numbers.

Order is Important

Scaffolding Task

Large Group

Using a number line for addition and subtraction

MCC.2OA.2

Gain an understanding of the concept and application of the commutative property of addition.

Different Paths,

Same Destination

Constructing Task

Large Group, Partners

Using a 99 chart

MGSE2.OA.2

MGSE2.NBT.5

Use benchmark numbers to add and subtract ones and tens using a 99 or hundred chart.

Number Destinations

Practice Task

Individual

Using a 99 chart

MGSE2.OA.1

MGSE2.OA.2

MGSE2.NBT.5

Apply understanding of ten and multiples of 10 being an appropriate benchmark number to design a set of directions from a start number of choice to a specific number destination on a number chart.

Our Number Riddles/My Number Riddle

Constructing Task

Large group, Partners Practice Task

Individual

Using a 99 chart

MGSE2.OA.1

MGSE2.OA.2

MGSE2.NBT.5

Correctly describe a number through written clues.

Building/Busting Towers of 10

Constructing Task

Partners

Represent numbers using models, diagrams, and number sentences

MGSE2.OA.1

MGSE2.OA.2

MGSE2.NBT.5

Experience the action of composing and decomposing groups of ten and how these two actions are opposite (inverse) operations.

Story Problems

Constructing Task

Individual

Representing numbers,

Addition and Subtraction

MGSE2.OA.1

MGSE2.OA.2

MGSE2.NBT.5

MGSE2.MD.8

Solve and verify story problems using various strategies.

Roll Away

Practice Task

Individual

Estimation, Mental math strategies

MGSE2.OA.2

MGSE2.NBT.5

Practice addition and mental math estimation skills.

Mental Math

Constructing Task

Large Group, Small Groups

Estimation, Mental math strategies

MGSE2.OA.2

MGSE2.NBT.5

Develop efficient ways to group numbers and or develop compensation strategies for mental addition and subtraction; and practice explaining strategies.

Take 100

Practice Task

Large Group

Addition to 100

MGSE2.OA.2

MGSE2.NBT.5

Develop mental math strategies in a game adding 2-dgit numbers to 100.

Multi-digit Addition

Scaffolding Task

Individual

Multi-digit addition with regrouping

MGSE2.OA.1

MGSE2.OA.2

MGSE2.NBT.5

Use mental math strategies and/or manipulatives to solve 2-digit story problems.

Addition Strategies

Constructing Task


Multi-digit addition with regrouping

MGSE2.OA.1

MGSE2.OA.2

MGSE2.NBT.5

Use mental math strategies and/or manipulatives to solve 2-digit story problems.

Caterpillars and Leaves (FAL)

MGSE2.OA.1

MGSE2.OA.2

Students demonstrate developing understanding of addition and subtraction

Sale Flyer Shopping

Constructing Task

Individual

Addition with money

MGSE2.OA.1

MGSE2.OA.2

MGSE2.NBT.5

MGSE2.MD.8

Further develop understanding of the concept of addition while solving problems by adding prices of purchases using only coin amounts.

Grocery Store Math

Practice Task

Large group, Partners

Modeling addition with money

MGSE2.OA.1

MGSE2.OA.2

MGSE2.NBT.5

MGSE2.MD.8

Add dollar amounts of purchases using the $ symbol correctly.

Subtraction: Modeling w/ regrouping

Scaffolding Task

Large group, Partners

Multi-digit subtraction with regrouping

MGSE2.OA.1

MGSE2.OA.2

MGSE2.NBT.5

Apply knowledge of addition and subtraction as inverse operations to develop a personal strategy for regrouping/trading in subtraction problems.

Subtraction Story Problems

Practice Task Individual

Multi-digit subtraction with regrouping

MGSE2.OA.1

MGSE2.OA.2

MGSE2.NBT.5

Independently solve subtraction story problems, defend his/her answers and strategies; show how addition can be used to check subtraction (inverse operations)

Menu Math

Practice Task

Individual

Addition and subtraction with money

MGSE2.OA.1

MGSE2.OA.2

MGSE2.NBT.5

MGSE2.MD.8

Create a menu with prices, estimate and add prices of purchases and determine appropriate change.

Counting Mice

Constructing Task


Multi-digit addition and subtraction

MGSE2.OA.1

MGSE2.OA.2

MGSE2.NBT.5

Use and defend various strategies to solve and check story problems.

Every Picture Tells a Story

Practice Task

Individual

Multi-digit addition and subtraction

MGSE2.OA.1

MGSE2.OA.2

MGSE2.NBT.5

Apply their knowledge of addition and subtraction to create a story problem from a picture and write an equation to match.

Planning a Field Trip

Culminating Task

Individual

Summative Assessment

MGSE2.OA.1

MGSE2.OA.2

MGSE2.NBT.5

MGSE2.MD.8

MGSE2.MD.10

Apply knowledge of all standards covered in this unit by planning a field trip. Students will also graph their data with teacher assistance.

If you would like further information about this unit, please view the Unit 2 Webinar on the K-5 math wiki.

INTERVENTION TABLE

The Intervention Table below provides links to interventions specific to this unit. The interventions support students and teachers in filling foundational gaps revealed as students work through the unit. All listed interventions are from New Zealands Numeracy Project.

Cluster of Standards

Name of Intervention

Snapshot of summary or

Student I can statement. . .

Materials Master

Numbers and Operations in

Base Ten

Use place value understanding and properties of operations to add and subtract

MGSE2.NBT.5

Counting

Say the forward and backward number word sequence in the range 0-100, starting and ending with any number.

Make 100

Develop an understanding of the quantity 100 and understand the relationship between 100 and 10

Number Line Flips

Order numbers in the range 0100

MM 4-2

More Ones and Tens

Solve addition and subtraction problems with decade numbers by counting

MM 4-14

Operation in Algebraic Thinking

Represent and solve problems involving addition and subtraction

Add and subtract within 20

MGSE2.OA.1

MGSE2.OA.2

Adding and Subtracting with Counters

Solve addition problems to 20 by joining sets and counting all the objects.

What's Hidden

Solve subtraction problems from 20 by counting all the objects in their head.

Imaging with Tens Frames

Solve addition and subtraction problems to 20 by counting all the objects in their head.

MM 4-6

Measurement and Data

Work with time and money

MGSE2.MD.8

Counting

Say the forwards and backwards skipcounting sequences in the range 0100 for twos, fives, and tens

MM 4-2

3 ACT TASK: Got Milk? Back to Task Table

APPROXIMATE TIME: ONE CLASS SESSION

In this lesson, students will use a picture to create a story problem involving addition and/or subtraction.


MGSE2.OA.1 Use addition and subtraction within 100 to solve one and two step word problems by using drawings and equations with a symbol for the unknown number to represent the problem. Problems include contexts that involve adding to, taking from, putting together/taking apart (part/part/whole) and comparing with unknowns in all positions.

MGSE2.NBT.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction

STANDARDS FOR MATHEMATICAL PRACTICE

1. Make sense of problems and persevere in solving them. Students are required to figure out a question to work through, the information they need to solve the problem, and then persevere until solving it.

2. Reason abstractly and quantitatively. Students are asked to make an estimate both high and low, as well as plot it on a number line.

3. Construct viable arguments and critique the reasoning of others. Students are given the chance to share and critique the questions and strategies of fellow classmates.

4. Model with mathematics. Students will use the information given to develop a mathematical model to solve their problems.

5. Use appropriate tools strategically. Students can use Base 10 blocks to aid in addition/subtraction strategies.

6. Attend to precision. Students will use clear and precise language when discussing their strategies and sharing their solutions with others.

7. Look for and make use of structure. Students will use their understanding of place value to help them add and subtract two digit numbers.

ESSENTIAL QUESTIONS (These questions will help support your lesson closing.)

How do we solve problems in different ways?

How can we show/represent problems in different ways?

How can different combinations of numbers and operations be used to represent the same

quantity?

How are addition and subtraction alike and how are they different?

MATERIALS

Cow Picture

Student Handout

GROUPING

Individual/Partner Task

TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION

In this task, students will view the picture and tell what they noticed. Next, they will be asked to discuss what they wonder about or are curious about. These questions will be recorded on a class chart or on the board and on the student recording sheet. Students will then use mathematics to answer their own questions. Students will be given information to solve the problem based on need. When they realize they dont have the information they need, and ask for it, it will be given to them.

Background Knowledge:

This task follows the 3-Act Math Task format originally developed by Dan Meyer. More information on this type of task may be found at http://blog.mrmeyer.com/category/3acts/. A Three-Act Task is a whole-group mathematics task consisting of 3 distinct parts: an engaging and perplexing Act One, an information and solution seeking Act Two, and a solution discussion and solution revealing Act Three. More information along with guidelines for 3-Act Tasks may be found in the Guide to Three-Act Tasks on georgiastandards.org and the K-5 GSE Mathematics Wiki.

In this task students will be shown a picture of a herd of cattle and asked what they wonder about. The main purpose of this task is to get students to practice adding or subtracting two digit numbers. Students can approach this in various ways based on the questions they ask. It is imperative that the teacher only provides information that will force a student to use addition or subtraction and not give too much information. Example: If a students question is, How many black cows are there? the student needs to ask how many total cows there are and how many brown cows there are in order to use subtraction to solve the problem. Remember to only give information if it is asked for.

Students should be able to explain and discuss their strategies for answering their main questions to the picture posed in Act 1. They should also be able to picture the story in their minds including the objects and actions in the story. They should solve the problems using pictures, words, and numbers. They should act out the story to make sure pictures, words, and numbers that were used make sense.

COMMON MISCONCEPTIONS

Children must come to realize that errors provide opportunities for growth as they are uncovered and explained. Trust must be established with an understanding that it is okay to make mistakes. Without this trust, many ideas will never be shared. (Van de Walle, Lovin, Karp, Bay-Williams, Teaching Student-Centered Mathematics, Developmentally Appropriate Instruction for Grades Pre-K-2, 2014, pg. 11)

It is important that students avoid using key words to solve problems. The goal is for students to make sense of the problem and understand what it is asking them to do, rather than search for tricks and/or guess at the operation needed to solve the problem.

Task Directions:

Act I Whole Group - Pose the conflict and introduce students to the scenario by showing Act I picture.

(Dan Meyer http://blog.mrmeyer.com/2011/the-three-acts-of-a-mathematical-story/)

Introduce the central conflict of your story/task clearly, visually, viscerally, using as few words as possible.

1. Show picture of cows to students.

2. Ask students what they noticed in the picture. The teacher records this information.

3. Ask students what they wonder about and what questions they have about what they saw. Students should share with each other first, and then the teacher records these questions (think-pair-share). The teacher may need to guide students so that the questions generated are math-related.

4. Ask students to estimate answers to their questions (think-pair-share). Students will write their best estimate, then write two more estimates one that is too low and one that is too high so that they establish a range in which the solution should occur.

Anticipated questions students may ask and wish to answer:

How many cows are there?

How many cows are brown? *

How many cows are black? *

Are there more brown cows or black cows? How many more?*

*Main question(s) to be investigated

Act 2 Student Exploration - Provide additional information as students work toward solutions to their questions. (Dan Meyer http://blog.mrmeyer.com/2011/the-three-acts-of-a-mathematical-story/)

The protagonist/student overcomes obstacles, looks for resources, and develops new tools.

During Act 2, students use the main question(s) from Act 1 and decide on the facts, tools, and other information needed to answer the question(s). When students decide what they need to solve the problem, they should ask for those things. It is pivotal to the problem-solving process that students decide what is needed without being given the information up front. Some groups might need scaffolds to guide them. The teacher should question groups who seem to be moving in the wrong direction or might not know where to begin.

The teacher provides guidance as needed during this phase. Some groups might need scaffolds to guide them. The teacher should question groups who seem to be moving in the wrong direction or might not know where to begin. Questioning is an effective strategy that can be used, with questions such as:

What is the problem you are trying to solve?

What do you think affects the situation?

Can you explain what youve done so far?

What strategies are you using?

What assumptions are you making?

What tools or models may help you?

Why is that true?

Does that make sense?

Additional Information for Act 2

There are 42 cows.

There are 20 black cows.

There are 22 brown cows.

Important note: Although students will only investigate the main question(s) for this task, it is important for the teacher to not ignore student generated questions. Additional questions may be answered after theyve found a solution to the main question, or as homework or extra projects.

Act 3 Whole Group Share solutions and strategies.

Students to present their solutions and strategies and compare them.

Reveal the solution.

Lead discussion to compare these, asking questions such as:

How reasonable was your estimate?

Which strategy was most efficient?

Can you think of another method that might have worked?

What might you do differently next time?

Act 4, The Sequel - The goals of the sequel task are to a) challenge students who finished quickly so b) I can help students who need my help. It can't feel like punishment for good work. It can't seem like drudgery. It has to entice and activate the imagination. Dan Meyer http://blog.mrmeyer.com/2013/teaching-with-three-act-tasks-act-three-sequel/

Challenge students to answer one of the student generated questions.

Challenge students to figure out how many ears are on the brown/black cows all together.

Challenge students to explain what would happen if 6 brown cows left, and 9 black cows joined the herd.

FORMATIVE ASSESSMENT QUESTIONS (These questions should be used during student worktime to facilitate conversations.)

How reasonable was your estimate?

What might you do differently next time?

What worked well for you this time?

What model did you use?

What organizational strategies did you use?

Did you group the cows to determine how many there are? If so, how did you group them? Is there a different way you could group them?

DIFFERENTIATION

Extension

Have students write an addition and a subtraction story problem involving the picture and solve it.

How many legs are on all the brown cows together? Black cows?

Intervention

Give students Base 10 blocks or another manipulative to help compute the problem.

Intervention Table

Name ______________________________________ Date ____________________

What problem are you trying to figure out?

What information do you already know?

What information do you need to solve the problem?

Make an estimate.

Write an estimate thats too low.

Write an estimate thats too high.

Show your estimates on a number line:

Show your work.

What is your conclusion? Show the answer using pictures, words, and numbers.

SCAFFOLDING TASK: Incredible Equations Back to Task Table

Approximately 1 Day (May be added to daily routines such as Number Corner)

In this task, the student will connect quantities with a position on a number line, express numbers in standard, written, and expanded form, write story problems, and generate fact families.



STANDARDS FOR MATHEMATICAL PRACTICE (SMP)

Although all standards for mathematical practice should be applied regularly, this task lends itself to the standards below:

1. Make sense of problems and persevere in solving them.

3. Construct viable arguments and critique the reasoning of others.

Independently written story problems are shared and explained with classmates.

4. Model with mathematics.

Students represent numbers in a variety of ways.

6. Attend to precision.

BACKGROUND KNOWLEDGE

This standard calls for students to add and subtract numbers within 100 in the context of one and two step word problems. Students should have ample experiences working on various types of problems that have unknowns in all positions using drawings, objects and equations. Students can use place value blocks or hundreds charts, or create drawings of place value blocks or number lines to support their work.

This task serves as a scaffolding task; however, it could easily become part of your daily or weekly routine. The suggested conversations in this type of task help students develop fluency with numbers by providing them continued experiences with multiple ways of composing and decomposing numbers. These types of experiences and conversations about numbers are also important for further development of the understanding of the magnitude of a given number as well as its relationship to other numbers.

ESSENTIAL QUESTIONS These questions may be used to guide the closing of the lesson.


How are the operations of addition and subtraction alike and different?

MATERIALS

Paper and pencil (keeping a recording of continued experiences with this type of task could be done in a math journal)

GROUPING

Large group; small group; and/or individual

TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION

Part I

Select a number to be the number of the day. To begin with you can choose any number between 0-20. This can be done by selecting one randomly from a jar. Other examples of how to come up with the number: use the date, use the number of students present in class that day, the number of students wearing a particular color, the number of students who are eating school lunch, the temperature outside, etc.

Challenge students to represent it in as many ways as possible place value, word form, expanded notation, addition and subtraction equations; money, etc.

Once students have had 3-5 minutes to write down their ideas have them share their work with a neighbor. Encourage them to notice both similarities and differences in what they did and what their partner did. Ask them to be ready to share one of the ways their neighbor expressed the number that was different from their own.

Then bring the class back together to discuss and share their work.

Ask questions like:

Where does this number live on the number line?

What numbers live right next door to this number?

What is 1 more? 1 less? 10 more? 10 less?

What kind of equations did you write that generated this amount?

How were the equations different? How were they the same?

Did you use any kind of strategy for expressing this amount? How was your neighbors work similar to your work? How was it different?

Do you think this number could be shared equally between 2 people? Why or why not?

Part II

Have students either work with a partner or independently to write a short story problem that matches one of the equations that either they came up with on their own or noticed during pair/share time or one that was shared during the class discussion time.

Have the students read their stories to a partner (or small group or whole class) and have those listening to the problem and identify whether it is an addition or a subtraction problem. Discuss with the students how they decided that the problem was addition or subtraction.

Part III

Select one of the equations the students wrote and have them generate other equations using those same numbers (i.e. creating fact families). Provide an example if students are having difficulty recognizing/remembering the idea of fact family relationships. If this is something that seems foreign to a student, see intervention suggestion below.

COMMON MISCONCEPTIONS:

When discussing fact family relationships, students may incorrectly order the subtraction equations. Example: Student writes 3 + 2 = 5 and 2 + 3 = 5 for addition; but writes

3 5 = 2 and 2 5 = 3 for subtraction.

Students do not make connections to inverse operations and revert to use of fingers for subtraction.

FORMATIVE ASSESSMENT QUESTIONS (Use these questions to facilitate student thinking during worktime.)

Where does this number live on the number line?

What numbers live right next door to this number?

What is 1 more? 1 less? 10 more? 10 less?

What kind of equations did you write that generated this amount?

How were the equations different? How were they the same?

Did you use any kind of strategy for expressing this amount? How was your neighbors work similar to your work? How was it different?

Do you think this number could be shared equally between 2 people? Why or why not?

DIFFERENTIATION

Extension

Introduce the idea of a T-chart and ask students to think about how they could use one to generate multiple equations for a given number.

Invite students to choose a number, create their own number line, write 2 or 3 questions, and share with classmates.

Intervention

Allow students to use manipulatives such as connecting cubes, buttons, or chips to make the given amount. Then show them how to separate the items into two different groups and how they can write an equation to represent what they just did. Using those same groupings also discuss with the student how to create a visual of what a fact family looks like. Make sure students can model both addition and subtraction examples with the manipulatives before having them record their work on paper.

The website http://nlvm.usu.edu/en/nav/frames_asid_156_g_1_t_1.html, gives practice with addition and subtraction. The students may work as a large group or with partners as the number of computers allows. Begin with addition problems, then move to subtraction problems. The students should draw what they think the number lines will look like and discuss their number lines and their answers. Then have the computer show the correct answer. This could even be used as a game or competition with prizes appropriate for your class. The game could be repeated on multiple days to reinforce this concept. This website requires the use of Java.

Intervention Table

SCAFFOLDING TASK: Order is Important Back to Task Table

Approximately 2 Days

In this task the student will use manipulatives to model addition and subtraction stories, explore the commutative property of addition, and continue to develop his/her understanding of fact families






3. Construct viable arguments and critique the reasoning of others.

Students are asked to prove addition and subtraction solutions as they use

manipulatives.

4. Model with mathematics.

Students represent fact families with manipulatives and Fact Family Houses.


7. Look for and make use of structure.

Students make connections between addition and subtraction through fact families.


This task serves as a scaffolding task; however, similar work was done in first grade. You may feel like your students can accomplish all three parts of this task in one day or perhaps you may use this task as an intervention only for those students who are still struggling with basic understanding of addition and subtraction. The suggested conversations in this task are important for further developing the concept of addition and subtraction.

The focus of the task is on the CONCEPT and application of the commutative propertynot on the definition or words. Students are also revisiting fact families as well as understating what strategies to use when solving an equation. Students are usually proficient when they focus on a strategy relevant to one particular fact. When these facts are mixed with others (as in a fact family), students may revert to counting as a strategy and ignore the efficient strategies they learned. Provide a list of facts from two or more strategies and ask students to name a strategy that would work for that fact. Students need to explain why they chose that strategy, then show how to use it to solve the equation. Additionally, they need to understand how to attach an equation to a particular story problem.

Be sure to discuss the concept of 0 and what happens when it is added or subtracted from a given amount. Students may over generalize the idea that answers to addition problems must be bigger. Adding 0 to any number results in a sum that is equal to that number. Provide word problems involving 0 and have students model them, using drawings with an empty space for 0.

Special comment- When students are building the towers make sure they keep the colors snapped together and they do not mix the colors up making a color pattern. Keeping the colors together will allow for them to visually see how numbers can be added and taken away for the connection to fact families. Repeat this task as a class until students are able to answer these questions and explain their reasoning.

ESSENTIAL QUESTIONS

These questions can help you facilitate the closing of the lesson.


Is order important? Why or why not?

MATERIALS

Connecting cubes (Individual bags of 18 cubes 9 each of 2 different colors)

Large number line (using masking tape or other materials)

Ready, Set, Hop by Stuart Murphy or similar book

GROUPING

Large group, small group


Part I

Gather students in meeting area. Read, Ready Set Hop, by Stuart Murphy or a similar book. Encourage students to act out the story using a large number line on the floor. Ask students, What happens if the frog hops forward? Backwards? Be aware this may be some students first experience using a number line to locate and compare numbers.

Teacher models addition and subtraction stories using 2 towers of different colored cubes. Have students build two towers of 9 cubes each each tower should contain 9 cubes of the same color but the two towers should be two different colors. Call out any addition problem using single digits such as 7+4. Have students build 7+4 using their towers (represent 7 in one color and 4 in a different color). Discuss that the sum of 7+4 is 11. Ask students how they can prove this? (by counting the cubes). What if we wanted to know what 4+7 was? What could we do? Take ideas from the studentsif no one mentions it, ask if they could flip their tower. Would that show 4+7? Yes. What is the sum of 4+7? It is still 1l. Write these problems using a number sentence on the board.

7 + 4 = 11

4 + 7 = 11

Have students model the action of adding these amounts together on the number line.

Make sure to use vocabulary like putting together, combining, adding to, and joining when discussing the action of addition with students.

While students still have the tower built, tell them to look at the 11 they have in front of them and take 4 of one color away They will see the difference is 7 write that number sentence on the board 11 4 = 7. Say, We had 11 and removed 4, now what do we have? How do you know? Then snap the 4 cubes back on the 7 and ask, What is our total? Why did it change?

Take off 7 to model 11-7. Ask, How is this different from what we just did? The difference is 4. Write that on the board.

11 4 = 7

11 7 = 4

Have students model the action of subtracting these amounts on the number line.

Make sure to use vocabulary like comparing, separating, removing, taking away, and counting back when discussing the action of subtraction with students.

Ask students to look at the numbers on the board and see what all problems have in common. Allow time for discussion. After discussions, test some more numbers to see if this is true with other problems. Use the related terms, facts, and fact family to discuss this concept. Students should recall from previous work in 1st grade that they are creating fact families. See interventions if students are having difficulty with this part of the task.

Make sure to encourage discussion about addition and subtraction being inverse operations (opposite of one another). The action of addition generates a total whereas in the action of subtraction a total is already known.

Part II

Say to students, Today you will determine if the order of the numbers affects the solution for addition and subtraction. Provide each student with a set of Unifix cubes. Say to students, Use your cubes to represent the numbers 12 and 3 (12 first then 3). Allow students time to create towers, then say Find the sum by linking the cubes and discuss your solution with the class. Watch as your teacher writes a number sentence to represent the sum you found. Now arrange your cubes so that they represent 3 and 12 (3 first then 12). Find the sum by linking the cubes and discuss your solution with the class. Watch as your teacher writes a number sentence to represent the sum you found. Did you get the same result both times? Participate in the class discussion about what happens to the sum of two numbers when you change the order of the numbers. Now, represent the number 12 using the cubes. Remove three of the cubes and discuss your result with the class. Watch as your teacher writes a number sentence to represent the difference you found. Now arrange your cubes so that they represent the number 3. Can you remove 12 cubes? Discuss your answer with the class. Did you get the same result both times?

Participate in the class discussion about what happens to the difference between two numbers when you change the order of the numbers. As in Part I of this task, continue to have discussions with students that encourage them to notice the relationship between addition and subtraction. Invite continued discussion about addition and subtraction being inverse operations (opposite of one another) and talk about the action of addition generates a total whereas in the action of subtraction a total is already known.

Part III

Make a simple pattern of a house (triangle on top of a square) for your students to trace and cut out of construction paper. They may choose whatever color they wish. They will need to have 2 copies that they have traced to serve as the front and back covers of their fact family book. They may design the cover of their book but will need to label it [Students name]s Fact Family House. You may want to cut out white paper in the shape of the house for the pages inside of the book. Each student will need at least 5 white pages for the inside of their book. Have the students choose 4 or 5 numbers between 10 and 18 and write one of the numbers on the top of each roof. You may wish to demonstrate each of these steps as the children begin so they can see what you are doing. The number the children choose will serve as the largest member of each fact family, and there will be a different fact family on each page of the book.

For example, say you chose 15. You would write 15 at the top of your roof on one of your white pages of paper. In each of the other corners of the roof, you would write a number smaller than 15 so that the two numbers have a sum of 15. If you chose 8, then 7 would be your other number and the roof would look like this.

15

8

7

Next, students will list the addition and subtraction fact family members in the box under the roof. Then they will use one of the fact family members to write a story problem at the bottom of the page. The reader will have to decide which fact was used to write the story. Students should write the answer on the back of the page. They will continue doing this with the other numbers they have chosen until they have 4 5 pages with a different fact family on each page.

15

8

7

15 children went swimming in the ocean. 8 children had blue rafts. The rest had red rafts. How many children had red rafts?

Answer: back of the page

8 + 7 = 15 15 8 = 7

7 + 8 = 15 15 7 = 8

FORMATIVE ASSESSMENT QUESTIONS These questions may be used to facilitate the student thinking during worktime.

What patterns do you see when you create a fact family?

How is a number affected when we add or take away zero?

Will changing the order of the numbers in an equation change the result, or answer, to the problem? If not why, if so when?

How can you represent the same amount using different combinations of numbers?

In what ways are addition and subtraction similar? In what ways are they different?

How do you know, or decide, what equation to write to represent a story problem?

What is another way to solve this problem?

DIFFERENTIATION

Extension

Use a deck of cards and dominoes to write fact families.

Intervention

http://illuminations.nctm.org/LessonDetail.aspx?ID=L57

In this lesson, the relationship of subtraction to addition is introduced with a book and with dominoes.

Provide students with multiple practices using manipulatives (bears, cubes, craft sticks, etc) to show the relationship between addition and subtraction fact families.

Intervention Table

CONSTRUCTING TASK: Different Paths, Same Destination Back to Task Table

Approximately 2-3 Days Due to the high usage of a 0-99 chart in grades K-1, consider using a 1-120 chart (attached) in your class. Later in the year, consider using a 100-200 chart for activities such as this.

In this task, the student will practice using benchmark numbers as he/she adds and subtracts 1s and 10s using a 0- 99 or other appropriate number chart.



MGSE2.NBT.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.




5. Use appropriate tools strategically.

Student use of 99 chart.


7. Look for and make use of structure.

Students look for patterns when adding and subtracting 1s and 10s on the 99 chart.


This game will address many different standards and involve listening and problem-solving strategies.

Students are usually proficient when they focus on a strategy relevant to particular facts. When these facts are mixed with others, students may revert to counting as a strategy and ignore the efficient strategies they have previously learned. Providing a list (perhaps on chart paper) of facts that includes strategies the students use for solving the facts can be a helpful reference for students. Make sure before posting any strategies that students explain why they chose that particular strategy for that particular fact and have shown show (explain) how to use it.

ESSENTIAL QUESTIONS

These questions may be used to facilitate the closing of the lesson.


How can we use skip counting to help us solve problems?

How does using ten as a benchmark number help us add or subtract?

MATERIALS

99 chart per student

Class 99 Chart

Paper/math journals

Transparent counters or highlighters

GROUPING


NUMBER TALKS

This task will provide a good opportunity to engage students in a number talk about why 10 is used as a benchmark number. For example, ask your students to solve the following equations:

15 + 10

15 + 11

32 + 10

32 + 11

54 + 10

54 + 11

Ask students to share how they used 10 to help them solve the equations quickly.

(For more information, refer to Sherry Parrish, Number Talks, and grades K-5.)


Part I

This task may provide teachers with the occasion to emphasize and model the importance of good listening skills. Explain to the children they will need to be especially good listeners in order to end up on the correct number.

Gather students in the meeting area. Display the class the 99 chart. Give each student a 99 chart. Select a starting number. Have students place a transparent counter on it or highlight it. Give students directions one at a time using the terms add 10, subtract 10, add 1, subtract 1, 10 more, 10 less, 1 more, and 1 less. After each clue, give students the opportunity to count up using their chart, if they need to and then have students move their transparent counter to the new number. Model this with the class, using only 3 or 4 directions. When the last direction has been given, ask students what number their transparent counter is on.

Sample direction set:

Place your counter on 16.

Add 10. (students should move their counter to 26)

Subtract 1. (students should move their counter to 25)

Move ahead 10 more. (students should move their counter to 35)

What number is the counter covering? (35)

Repeat this activity several times as a class making sure to vary directions to include subtracting, moving back 1 or 10, 10 more, 10 less etc. Once students are comfortable with following the given directions, proceed to part II of the task.

Part II

Tell the students the game directions have now changed. Explain to the students that you need their help to create the directions to get to the number 45 from the number 14. Use the large class 99 chart to model the directions offered by students. Ask students to suggest directions. Possible scenario may include Add 10 to 14. Now, where are we? (24) Add another group of ten. Where are we now? (34) Add 10 once more. (44) We are almost there, what should I add now? (1 more) Where did we end? (45)

Students directions will vary, ask students to share. Encourage conversations about the difference in addition strategies presented. It is important to discuss how adding and subtracting 10 is more efficient. This also allows students to practice using 10 as a benchmark number. Helping students to see that adding 12 done faster by adding 10 and then 2 more. Working with groups of 10 in this task gives students more practice with understanding benchmarks of 10.

Continue activity with several classroom examples until students appear comfortable with creating directions. Include examples with numbers that have a larger starting point than ending point, so that subtraction is involved.

Allow students to work with a partner to create their own set of directions for a specific number. The teacher will provide the ending point, but will allow students to select their own starting point. For instance, 27 may be the end point the teacher designates. One set of partners may choose to start at 48 and another at 7; however, they will all end at 27. Allow time for several partners to share their different pathways to 27. Using their 99 chart and their math journals, have students record their directions to 27. Make comments about various ways to get to the number 27, encouraging students to use benchmark numbers to navigate the numbers.

Part III

Allow students to select any number they choose as their final destination. Then instruct the students to create 3 different paths to the same destination (same number). Also instruct students to include subtraction in at least one of the paths.

FORMATIVE ASSESSMENT QUESTIONS These questions may be used to facilitate student thinking during the worktime portion of the task.

What happens to a number when we add ten to it? When we subtract ten from it?

Are you counting by ones when you add on a ten? Why?

What does it mean to skip count by ten? Why would we want to do this?

How do you think adding or subtracting by twenty would relate to adding and subtracting by ten?

DIFFERENTIATION

Extension

Play the I Have, Who Has? games. Examples and direction cards are available at http://math.about.com/od/mathlessonplans/ss/ihave.htm These games can be printed on cardstock and laminated for extended use.

Encourage students to use 2 or more subtractions of 1s or 10s in their paths.

Intervention

Teacher can select numbers which would allow students to focus on using directions, I am 1 or 10 less than_____, and I am 1 or 10 more than____. What is the number? This could be done with a sentence frame for students.

Intervention Table

Name ___________________________ Date______________

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

99 Chart

1-120 Chart

1

2

3

4

5

6

7

8

9

10

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